This paper is organized as follows: we start our study focusing in discrete time dynamical systems (Sections -REF ). In Section  we recall some notions of dichotomy and highlight some relations between them. Moreover, we introduce and study a notion of dichotomy spectrum for systems admitting a nonuniform polynomial dichotomy. In Section  we present results about \(C^1\) , differentiable (at 0) and Hölder linearization under the assumption of nonuniform exponential dichotomy on the half line. Section  is devoted to establish \(C^1\) , differentiable (at 0) and Hölder linearization results under the assumption of nonuniform polynomial dichotomy. At the end of Section , we explain how the results from the previous section can be extended to the infinite dimensional setting. Finally, in Section  we obtain continuous time versions of the results obtained in Section . Appendix A is devoted to strengthen the properties of the conjugacies given in [1]}, while in Appendix B we prove Proposition REF .
